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PPPC 4 DMν: a Poor Particle Physicist Cookbook for Neutrinos from Dark Matter
annihilations in the Sun
View the table of contents for this issue, or go to the journal homepage for more JCAP03(2014)053
(http://iopscience.iop.org/1475-7516/2014/03/053)
JCAP03(2014)053
ournal of Cosmology and Astroparticle Physics
An IOP and SISSA journal
J
PPPC 4 DMν: a Poor Particle
Physicist Cookbook for Neutrinos
from Dark Matter annihilations in
the Sun
Pietro Baratella,
aMarco Cirelli,
bAndi Hektor,
c,dJoosep Pata,
cMorten Piibeleht
cand Alessandro Strumia
c,eaScuola Normale Superiore and INFN,
Piazza dei Cavalieri 7, Pisa, 56126 Italy
bInstitut de Physique Th´eorique, CNRS URA 2306 & CEA-Saclay,
Gif-sur-Yvette, 91191 France
cNational Institute of Chemical Physics and Biophysics,
Ravala 10, Tallinn, Estonia
dHelsinki Institute of Physics,
P.O. Box 64, Helsinki, FI-00014 Finland
eDipartimento di Fisica dell’Universit`a di Pisa and INFN,
Largo Buonarroti 2, Pisa, Italy
E-mail: pietro.baratella@sissa.it,marco.cirelli@cea.fr,andi.hektor@cern.ch, joosep.pata@cern.ch,morten.piibeleht@cern.ch,alessandro.strumia@cern.ch Received January 8, 2014
Accepted March 18, 2014 Published March 27, 2014
Abstract. We provide ingredients and recipes for computing neutrino signals of TeV-scale
Dark Matter (DM) annihilations in the Sun. For each annihilation channel and DM mass we present the energy spectra of neutrinos at production, including: state-of-the-art energy losses of primary particles in solar matter, secondary neutrinos, electroweak radiation. We then present the spectra after propagation to the Earth, including (vacuum and matter) flavor oscillations and interactions in solar matter. We also provide a numerical computation of the capture rate of DM particles in the Sun. These results areavailable in numerical form.
Keywords: dark matter theory, ultra high energy photons and neutrinos
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Contents
1 Introduction 1
2 The DM annihilation rate 3
2.1 The DM capture rate 4
3 The neutrino energy spectra at production 7
3.1 Neutrino spectra with MonteCarlo codes 8
3.1.1 Pythia 8
3.1.2 Geant 9
3.1.3 Results and comparisons 10
3.2 Adding electroweak bremsstrahlung 11
3.3 Spectra at production: results 13
4 Neutrino propagation 13 4.1 Formalism 13 4.2 Transition probabilities 16 4.3 Earth crossing 18 5 Final result 19 6 Conclusions 19 1 Introduction
The Dark Matter (DM) particles that constitute the halo of the Milky Way have a small but finite probability of scattering with a nucleus of a massive celestial body (e.g. the Sun or the Earth) if their orbit passes through it. If their velocity after the scattering is smaller than the escape velocity from that body, they become gravitationally bound and start orbiting around the body. Upon additional scatterings, they eventually sink into the center of the body and accumulate, building up a local DM overdensity concentrated in a relatively small volume. There they annihilate into Standard Model particles, giving origin to fluxes of energetic neutrinos [1–4]. These neutrinos are the only species that can emerge, after experiencing oscillations and interactions in the dense matter of the astrophysical body. The detection of high-energy neutrinos from the center of the Sun or the Earth, on top of the much lower energy neutrino flux due to nuclear fusion or radioactive processes, would arguably constitute one of the best proverbial smoking guns for DM, as there are no known astrophysical processes able to mimic it (except possibly for the flux of neutrinos produced in the atmosphere of the Sun, which however are expected to have a different spectral shape [5,6]).
While the above idea is rather old, recently it has gained more interest since large neu-trino telescopes (Antares, Icecube and its planned extension Pingu, SuperKamiokande and its proposed upgrade HyperKamiokande, the future Km3Net) are reaching the sen-sitivity necessary to probe one of the most interesting portions of the parameter space, in competition with other DM search strategies.
Along the years there have been several computations of the ingredients necessary to predict the signal in a detector, in particular the DM capture rate [7–13] and the spectra of
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neutrinos [14–21]. In [18] a subset of the authors of this paper had computed the neutrino spectra: their production in solar matter and their propagation subject to interactions and flavor oscillations. Similar results, but in an event-based approach, had been subsequently obtained by Wimpsim [19].In this paper we improve the computation of the neutrino energy spectra with respect to previous computations and in particular to [18] in various ways:
• we consider a full list of two-body SM annihilation channels;
• we implement a better description of the energy losses of the primary particles in solar matter, including secondary neutrinos produced by scattered solar matter;
• we extend to low energy the neutrino spectra in order to include the large flux of neutrinos generated by stopped µ, π, K (an effect pointed out in [20,21]);
• we take into account EW bremsstrahlung;
• we update the neutrino oscillation parameters adding the recently discovered θ13≈ 8.8◦
and the two possible mass hierarchies of neutrinos (normal or inverted).
Let us now review the main steps of the computation and present the results we provide. The final energy spectrum of the neutrino flux at the detector location is written as
dΦν dEν = Γann 4πd2 dNν dEν (1.1)
where d is the Sun-Earth distance, Γannis the total DM annihilation rate in the Sun, dNν/dEν
is the energy spectrum of νe,µ,τ and ¯νe,µ,τ produced per DM annihilation after taking into
account all effects. The computation proceeds as follows:
1. The total rate of DM annihilations Γann in the Sun is computed in section 2 in terms
of the spin-independent and spin-dependent DM cross sections with nucleons.
2. In section 3we compute the energy spectra of ν and ¯ν at production for DM annihila-tions into
e+Le−L, e+Re−R, µ+Lµ−L, µ+Rµ−R, τL+τL−, τR+τR−, νeν¯e, νµν¯µ, ντν¯τ,
q ¯q, c¯c, b¯b, t¯t, γγ, gg, WL+WL−, WT+WT−, ZLZL, ZTZT, hh
(1.2)
where q = u, d, s denotes a light quark, the subscripts L, R on leptons denote Left or Right-handed polarizations, the subscript L, T on massive vectors denote Longitudinal or T ransverse polarization. h denote the higgs boson, for which we assume a mass of 125 GeV.
Decays, hadronization, energy losses in matter and secondary neutrinos generated by matter particles scattered by DM decay products (an effect ignored in previous com-putations) are computed running Geant4 [22,23] using the EU Baltic Grid facilities. As a check, the first three effects are also independently computed with Pythia [24], modified by us to include energy losses in matter.
3. As described in section 3.2, we correct the spectra at production including the dom-inant electroweak radiation effects (not included in MonteCarlo codes) enhanced by logarithms of MDM/MW, as described in [25]. EW corrections depend on the
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4. As described in section 4, we propagate the fluxes of ν and ¯ν at production, aroundthe center of the Sun, to the Earth taking into account oscillations, matter effects, absorption and regeneration from collisions with matter using the neutrino density matrix formalism as in [18].
The final results are presented and briefly discussed in section 5. In section 6 we conclude.
The main numerical outputs of the computation are given on the PPPC 4 DM ID website (‘DMν’ section).
2 The DM annihilation rate
In this section we compute the DM solar annihilation rate, which sets the overall normal-ization of the expected neutrino flux. We essentially revisit the calculations in [7–13]. For definiteness, we focus on the case of the Sun, although we mention in passing the modifica-tions needed for the Earth.
The number N of DM particles accumulated inside the Sun varies with time under the action of different competing processes: 1) DM particles are captured from the halo (N is thus increased by one unit) via the multiple scattering processes discussed above, 2) DM can pair annihilate (hence N decreases by two units) and 3) DM particles can be ejected (N decreases by one unit) by a hard scattering on a hot nucleus of the interior of the body, i.e. the inverse process with respect to capture. In formulæ:
dN
dt = Γcapt− 2Γann− Γevap (2.1)
where Γcapt is the capture rate, Γann is the DM annihilation rate and Γevap is the DM
evaporation (or ejection) rate.
The evaporation process is important only for DM lighter than a few GeV,1 so that we will neglect it for all practical purposes in the following.
The annihilation rate is proportional to N2: two DM particles annihilate (hence the square). It is given by Γann = 1 2 Z d3x n2(~x) hσvi = 1 2CannN 2, (2.2)
where hσvi is the usual annihilation cross section averaged over the initial state2 and n(~x) is the number density of DM particles at position ~x inside the Sun, such that the total number of DM particles is N =R d3x n(~x). After capture, subsequent scatterings thermalize the DM
1
This can be easily understood [7–13]: if one requires that the speed of a DM particle thermalized with the interior of the Sun vDM ∼p2T /MDM be smaller than the escape velocity from the center of the Sun v esc ' 1387 km/s, one obtains the condition MDM & 0.15 GeV. A more realistic computation takes into account that DM particles actually follow a (Maxwellian) velocity distribution and compares the time it takes to deplete the high-velocity tail (above the escape velocity) of such distribution with the age of the Sun, finding MDM& 5 GeV. Yet more refined calculations find typically MDM& 4 GeV [26].
2
If DM is a real particle (e.g. a Majorana fermion) this is the usual definition of σ and the factor 1/2 takes into account the symmetry of the initial state. If DM is a complex particle (e.g. a Dirac fermion) then n ≡ nDM+ nDM(here assumed to be equal) and the average over initial states is σ ≡
1
4(2σDM DM+ σDM DM+ σDMDM). In many models, only DM DM annihilations are present, so that σ = σDM DM/2.
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particles to the solar temperature T , such that their density n(~x) acquires the sphericallysymmetrical Boltzmann form
n(r) = n0 exp[−MDMφ(r)/T ] (2.3)
where n0 is the central DM number density and φ(r) =
Rr
0 dr GNM (r)/r
2 is the Newtonian
gravitational potential inside the Sun, written in terms of the solar mass M (r) enclosed within a sphere of radius r. Taking for simplicity the matter density in this volume to be constant and equal to the central density ρ , all the integrals can be explicitly evaluated .
One finds that DM particles are concentrated around the center of Sun,
n(r) = n0e−r 2/r2 DM, with rDM= 3T 2π GNρ MDM 1/2 ≈ 0.01 R r 100 GeV MDM . (2.4)
Within this approximation, one obtains from eq. (2.2)
Cann = hσvi
GNMDMρ
3 T
3/2
. (2.5)
Here ρ = 151 g/cm3 and T = 15.5 106 K are the density and the temperature of matter
around the center of the Sun. The same expression would hold for other astrophysical bodies, adapting these two quantities.
Neglecting Γevap and solving eq. (2.1) with respect to time one finds
Γann= Γcapt 2 tanh 2 t τ t τ ' Γcapt 2 (2.6)
where τ = 1/pΓcaptCann is a time-scale set by the competing processes of capture and
annihilation. At late times t τ one can approximate tanh(t/τ ) = 1. In the case of the Sun, the age of the body (∼4.5 Gyr) and the typical values of the parameters in τ indeed satisfy this condition (in the case of the Earth this is not generally the case). Therefore one attains the last equality of eq. (2.6). Physically, this means that the fast (compared to the age of the Sun) processes of capture and annihilation come to an equilibrium: any additional captured particle thermalizes and eventually is annihilated away.
2.1 The DM capture rate
As a consequence of the attained equilibrium, the computation of the annihilation rate Γann
crucially depends on the computation of the capture rate Γcapt, which acts as a bottleneck.
The computation of the latter proceeds by summing the contribution to capture of all the individual shells of matter located at position r within the massive body. The result is [7–13]
Γcapt= ρDM MDM X i σi Z R 0 dr 4πr2 ni(r) Z ∞ 0 dv 4πv2f (v) v2+ v esc2 v ℘i(v, v esc). (2.7) Its derivation is lengthy: we will only sketch it in the following by illustrating the individual pieces of this equation.
• ρDM/MDM corresponds just to the local number density of DM particles at the location
of the capturing body.
• The sum runs over all kinds of nuclei i with mass mi and number density ni(r), to be
integrated over the volume of the Sun. We take the standard solar elemental abundances from [27–30].3 The factor σi is the low-energy DM cross section on nucleus i, assumed
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to be isotropic. In terms of the standard Spin Independent and Spin Dependent cross sections normalized on a single nucleon at zero momentum transfer4 (denoted σSI andσSD) we take σH= σSI+σSDfor Hydrogen and σHe' 44σSI(mp+MDM)2/(mHe+MDM)2
for the spin-less Helium. Analogously, for heavier nuclei of mass number A one can take σi' A2σSI(mA/mp)2 (mp+ MDM)2/(mA+ MDM)2 (spin independent contribution) +
σi ' A2σSD(mp+ MDM)2/(mA+ MDM)2 (spin dependent contribution). This latter
SD formula should be complemented by weighting factors due to the spin content in the different nuclei. In practice, only Hydrogen is relevant for our case, so that a more careful evaluation is not necessary.
• f (v) is the angular average of the DM velocity distribution with respect to the solar
rest frame neglecting the gravitational attraction of the Sun. This quantity is connected to the f (v) with respect to the galactic rest frame as
f (v) = 1 2 Z +1 −1 dc f ( q v2+ v2 + 2vv c) (2.8)
in terms of the solar velocity v ≈ 233 km/s (here c is the cosine of the angle
be-tween v and v ). Observations and numerical simulations suggest that f (v) can be
parameterized as [31–36] f (v) = N exp v 2 esc− v2 kv2 0 − 1 k θ(vesc− v) (2.9) with normalisationR∞ 0 dv 4πv 2f (v) = 1 and parameters 220 km/s < v0 < 270 km/s, 450 km/s < vesc < 650 km/s, 1.5 < k < 3.5. (2.10) Here vesc is the escape velocity from the Galaxy (not to be confused with v esc, the
escape velocity from the Sun) at the location of the solar system. For k = 0, f (v) reduces to a Maxwell-Boltzmann distribution with a sharp cut-off at v < vesc: f (v) =
N × e−v2/v20θ(vesc − v). For k > 0 the cut-off gets smoothed. The resulting f (v) are normalised such that R∞
0 dv 4πv
2f
(v) = 1. The actual choice of the DM velocity
distribution turns out to have a rather limited impact, see e.g. [37] for a recent analysis. • The gravity of the Sun is taken into account by v esc(r), which is the escape velocity
(from the Sun) at radius r, such that v2+v esc2 is the squared velocity that a DM particle acquires when arriving at r from a very distant point. One here neglects the effect of other bodies in the solar system, which presumably is a good approximation [38–44]. • The probability that a DM particle, with velocity v far from the Sun, gets captured
when scattering on a nucleus located where the escape velocity is v esc is given in first
approximation by ℘i(v, v esc) = max 0,∆max− ∆min ∆max (2.11)
where ∆max= 4miMDM/(MDM+mi)2and ∆min = v2/(v2+v esc2 ) are the maximal and
minimal fractional energy loss ∆E/E that a particle can suffer in the scattering process, 4
We assume that they are equal for protons and neutrons and that they do not depend on the momentum transfer nor on the DM velocity.
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1 10 100 1000 104 1025 1026 1027 1028 1029 1030 DM mass MDMGeV Capture rate Gcapt Σp in 1 sec *pbSpin Independent DM capture rate in the Sun
He N O Ne Mg Si S Fe tot v0= 250 kmsec vesc= 550 kmsec k= 1 (a) 1 10 100 1000 104 1023 1024 1025 1026 1027 1028 1029 DM mass MDMin GeV Capture rate Gcapt Σp in 1 sec *pb
Spin Dependent DM capture rate in the Sun
tot N H v0= 250 kmsec vesc= 550 kmsec k= 1 (b)
Figure 1. DM capture rate in the Sun Γcapt of DM particles with mass MDM assuming a
Spin-Independent (left) or a Spin-Dependent (right) DM cross section on matter. We normalize assuming a cross section on protons σp = 1 pb and we adopt the indicated values for the parameters of the
galactic DM velocity distribution. We show the total as well as the contributions from the most relevant individual elements in the Sun. The dashed line is the analytical approximation valid when the DM is much heavier than the nuclei.
provided that it is captured. I.e., the probability is computed as the ratio of the size of the interval in energy losses leading to capture (∆Emin < ∆E < ∆Emax) relative to the
whole possible interval (0 < ∆E < ∆Emax), assuming a flat distribution of the
scatter-ing cross section in energy. In general, however, one needs to introduce the form fac-tors Fi(∆E) that take into account the nuclear response as function of the momentum
transfer. Explicitly, |Fi(∆E)|2 = e−∆E/E0, with E0 = 5/2mir2i for spin-independent
and E0 = 3/2miri2 for spin-dependent scattering (here ri ∼ p3/5 A1/3i 1.23 fm '
0.754 · 10−13cm (mi/GeV)1/3 is the effective radius of a nucleus with mass number Ai
and mass mi). The numerator of the ‘ratio of sizes’ becomes then an integral of the
form factor over the energy loss ∆E:
℘i(v, v esc) = 1
E ∆max
Z E ∆max
E ∆min
d(∆E) |Fi(∆E)|2, (2.12)
eq. (2.11) and (2.12) mean that the fraction of scatterings that lead to capture is largest for nuclei with mass mi comparable to the DM mass MDM (∆max is maximized) and
for DM particles that are slow (small v) and in the central regions of the body (large v esc).
Figure 1a shows the capture rate in the Sun having assumed a Spin Indipendent cross section σpSI= 1 pb on protons. One sees that several elements contribute figure1b shows the Spin Dependent capture rate, with the corresponding assumption σpSD = 1 pb on protons. Only Hydrogen matters for this kind of capture, with a very small contribution from Nitrogen.
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The dotted lines in figure1 are simple approximations valid in the limit of heavy DM, MDM mi. In such a limit DM can be captured only if it is very slow,vMDM< mi 2v esc
p
mi/MDM. (2.13)
Thereby the capture rate is proportional to 1/MDM2 and can be approximated as Γcapt MDM mi ' ρDM MDM2 4πf (0) X i miσiIi (2.14) where Ii= Z R 0 4πr2ni(r) " 1 2 E0i mi 2 − E0i mi e−2miv2 esc(r)/E0i E0i 2mi + v esc2 (r) # dr (2.15)
In the limit of negligible form factors, E0i mi, the term in square brackets simplifies
to v esc4 .
The integrals Ii are adimensional in natural units, and their values are given in table 1
for the main capturing elements. Inserting their values we find
Γcapt' 3.28 · 10−16 sec ρDM 0.3GeVcm3 ! 100 GeV MDM 2 270km sec veff 0 !3 1.8 · 1046σSD+ 2.16 · 1049σSI pb . (2.16) We here parameterised f (0) = 1/π1/2v0eff
3
, such that the parameter veff0 coincides with the parameter v0 for a Maxwellian distribution with no cut-off and neglecting solar motion.
We also provide a numerical function which gives the annihilation rate Γann = Γcapt/2
in terms of MDM, the standard DM cross sections on a single nucleon σSDand σSI(as defined
above) and the parameters of the velocity distribution v0, vesc, k.
3 The neutrino energy spectra at production
In this section we discuss the computation of the neutrino fluxes emerging from the DM annihilation process. Neutrinos are produced at several stages of the hadronic and leptonic cascades originating from the primary particles produced by annihilations, and these cascades develop within the dense matter of the astronomical body (the Sun). Such an environment has important consequences in determining the spectra. For example, some byproducts of the cascade can be absorbed by the surrounding matter; some others will be just ‘slowed down’; others are negligibly affected. It is therefore important to model in the best possible way all these effects. We adopted and compared two strategies:
1) As described in section 3.1.1, we modified the public MonteCarlo code Pythia to include the effects of cascading energy losses with matter and absorption. The disad-vantage is that in this way we cannot include the neutrinos emitted by matter particles scattered by products of DM annihilation.
2) As described in section 3.1.2, we run the Geant4 code, dedicated to particle interac-tions with matter. The disadvantage is that Geant4 is much more time-consuming than Pythia: to reach a reasonable statistics we employ the computational resources of the Baltic grid.
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SD capture H 1.60 · 1047 N 2.42 · 1043 SI capture He 2.00 · 1046 N 2.84 · 1043 O 7.34 · 1043 Ne 1.03 · 1043 Mg 3.01 · 1042 Si 1.98 · 1042 S 5.73 · 1041 Fe 8.87 · 1040Table 1. Value in natural units of Ii, for the main elements of the Sun.
In section3.1.3we compare the two results, interpret them and decide to adopt the Geant4 result. In section 3.2 we describe how we subsequently add electroweak bremsstrahlung corrections, which are not included in any Monte Carlo code.
3.1 Neutrino spectra with MonteCarlo codes 3.1.1 Pythia
We here describe how we modified the Pythia public code to include the effect of cascading in matter in an event-by-event basis. We improved with respect to our previous computation [18] where we had only included averaged energy losses in matter. Moreover, we here use Pythia version 8.176 (version 6.2 was used in [18]).
The process of particles with mass m and initial energy E0 m decaying with life-time
τ , while at the same time losing energy due to interactions with the surrounding matter, is described by a differential equation for their energy E and their number n:
dE dt = −α − βE, dn dt = − n τ m E; (3.1)
that is: the energy loss of particles in matter can by approximated as a constant term α plus a term β proportional to their energy, while their number n is just governed by τ (the factor m/E takes into account time dilatation). Therefore
dn dE = dn/dt dE/dt = nm τ E(α + βE) (3.2) is solved by n(E) = n(E0) β + α/E0 β + α/E m/τ α . (3.3)
We thereby instruct Pythia to perform the decay of particles by first replacing their initial energy E0 with a lower energy E randomly chosen according to the above distribution
pro-duced by energy losses, eq. (3.3), and then letting them decay. This equation takes different specific forms according to the particle considered, as we discuss now.
Dominant β: stopping of hadrons. The stopping of hadrons is well approximated by α = 0. In such a case eq. (3.3) simplifies to
E(t) = E0e−βt, n(E) n(E0) = exp Ecr E0 − Ecr E(t) = ex0−x (3.4)
where Ecr= m/βτ is a critical energy below which energy losses are negligible. The
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Sun one has Ecr ≈ 250 GeV for charmed hadrons and Ecr ≈ 470 GeV for b-hadrons [14–17].This improves over our previous computation [18] where we used a constant average energy at decay, given by hEi = Z E dn = Ecr Z ndE E = Ecr Z ∞ x0 ex0−xdx x . (3.5)
Neutrons and negatively charged pions need however a peculiar treatment: once stopped, instead of decaying, they are absorbed by matter. Neutrons are absorbed by protons, forming deuteron. The π− are quickly Coulomb-captured by heavy nuclei in the dense and hot solar core environment, forming pionium which typicallys decay into a neutron and photons [45]. Since no neutrinos are produced in final states, to implement this phenomenology we simply modified our Pythia code in such a way that neutrons and pions are removed and not decayed.
Dominant α: stopping of charged leptons. The energy loss of a charged lepton can often be approximated as an energy-independent term (i.e. taking β = 0), which in the center of the Sun equals to
dE
dt = −α ≈ −
0.8 1010GeV
sec . (3.6)
The general solution of eq. (3.3) simplifies then to
E(t) = E0− αt, n(E) = n(E0)
E E0
m/τ α
. (3.7)
The exponent m/τ α equals 1/176000 for muons (which therefore lose almost all their energy before decaying) and equals 660 for taus (which therefore decay before losing a significant fraction of their initial energy). The average energy at decay employed in our previous computation is hEi = Z E dn = E0 1 + τ α/m. (3.8) 3.1.2 Geant
Geant4 is a toolkit for the simulation of the passage of particles through matter [22,23], a very suitable tool for modeling cascades in the environment of the solar core. We follow the approach of [20,21]. Hadronization of the quarks and gluons produced by DM annihilations is performed by Pythia; the stable and metastable hadronization products (e, νe,µ,τ, µ,
KL0, π, K, n, p, ND,He) are injected into Geant4 which adds the effect of particle/matter
interactions. We model the matter around the solar core following the Solar Standard Model as discussed in section 2, but with a simplified chemical composition (we just include 74.7 % by mass of H and 25.3 % of He). In our Geant4 code, we consider a sphere with radius of 1 km, that is big enough for our purposes: we verified that only neutrinos from the cascades can reach its surface, while secondary products are contained. The passage of particles through matter is simulated using the QGSP BERT physics list, see [22,23] for further details. Notice that neutrinos have no matter interactions nor oscillations in the QGSP BERT physics model of Geant4 (we will indeed discuss in detail how to deal with these effects in the subsequent neutrino propagation in matter in section 4).
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10-5 10-4 10-3 10-2 10-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x Νefrom DM DM®bb, MDM= 100 GeV HaL DMΝ H2005L Pythia Geant 35 MeV 250 MeV (a) 10-5 10-4 10-3 10-2 10-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x Νefrom DM DM®bb, MDM= 1000 GeV HbL DMΝ H2005L Pythia Geant (b) 10-5 10-4 10-3 10-2 10-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x Νefrom DM DM®bb, MDM= 10000 GeV HcL Pythia Geant (c) 10-5 10-4 10-3 10-2 10-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x ΝΜfrom DM DM®bb, MDM= 100 GeV HdL DMΝ H2005 L Pythia Geant 29.8 MeV 240 MeV (d) 10-5 10-4 10-3 10-2 10-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x Νefrom DM DM®W+W-, MDM= 100 GeV HeL DMΝ H2005L Pythia Geant (e) 10-5 10-4 10-3 10-2 10-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x Νefrom DM DM®Τ+Τ-, MDM= 1000 GeV HfL DMΝ H2005L Pythia Geant (f)Figure 2. Comparison between the spectra obtained with Pythia and with Geant, for a few representative cases. See text for comments.
3.1.3 Results and comparisons
Figure 2 presents a few examples of neutrino spectra obtained with the two MonteCarlo codes. It also shows (when available) the spectra previously computed in [18].
In very general terms, moving from low to high x = E/MDM, the new spectra are
characterized by some very pronounced low energy humps or spikes (missing in [18]), an intermediate energy smooth shoulder and, for some channels, a high energy peak. These features are easily understood.
• The high energy peak occurs when DM annihilate into particles that directly decay into neutrinos. This is visible in figure 2as a slanted mountain top in the W+W− channel (panel (e)). A similar peak is of course present for the ZZ channel (not shown). For large DM masses, i.e. for large boosts of the primary W and Z, this feature smears into a smooth spectrum (see e.g. panel (a) of figure 3).
• The smooth component of the spectrum arises from the neutrinos produced in the cascading event by primary and secondary particles (hadrons and leptons), that lose energy and rapidly decay.
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• The low energy humps and spikes essentially arise from relatively long-lived particlesthat have been stopped in solar matter and then decay, essentially at rest. More precisely, recalling that n, π−, µ− and K−are mostly absorbed or captured by matter, the low energy neutrino peaks arise from the following processes:
– π+ → µ+ν
µ decays, which produce a monochromatic line in the νµ spectra, at
Eν = 29.8 MeV (visible in panel (d) of figure2). For numerical reasons we
artifi-cially broaden it.
– µ+→ ¯νµνee+ decays, which contribute to the νe, ¯νµenergy spectra producing the
typical three body decay bump with an end-point at mµ/2 ≈ 53 MeV and peaked
at E ≈ mµ/3 ≈ 35 MeV (visible in panel (a)).
– µ− → νµν¯ee− decays, which similarly contribute to the ¯νe, νµ energy spectra,
although the resulting bump is about two orders of magnitude less intense than the bump in νe, ¯νµ. Indeed π− are absorbed by matter before decaying into µ−.
– K+decays with 63% branching fraction into a monochromatic νµat Eν = 240 MeV
(see panel (d)). Three body decays have smaller branching ratios (5.1% BR into π0 e+ νe and 3.4% BR into π0 µ+ νµ) producing bumps below about mK/2 ≈
250 MeV (see e.g. panel (a)).
– K− get absorbed, synthesised by nuclei into a Λ which decays into nucleons and pions. Their rare free decays negligibly affect the neutrino spectra.
– Decays of KL0 into neutrinos are blocked by matter effects that break the quantum coherence between K0
S and KL0, such that KS0 are continuously regenerated and
fastly decay hadronically.
While these general features are present both in the Pythia and the Geant spectra, the results of the two MonteCarlos differ in the details.
• For hadronic channels (e.g. b¯b), Pythia systematically produces a softer high-energy spectrum than Geant, as clearly visible in panels (a), (b) and (c) (and (d) too). The difference is small for small DM masses and increases for large DM masses. For the leptonic channels, on the other hand, the two results agree (see e.g. panel (f)). This can probably be ascribed to a difference in the way Pythia treats the energy losses of the energetic hadrons in the cascade.
• Pythia systematically underestimates the low energy humps and spikes, as it is clearly visible for instance in panel (f).5 This is expected since, as we anticipated, Pythia includes only the neutrinos produced by the decays of pions, kaons and muons in the DM-originated shower. Geant, instead, follows the fate of all the particles produced in the scattered matter, including therefore additional pions, kaons and muons that then decay into neutrinos.
Based on the considerations above, we adopt the Geant spectra. 3.2 Adding electroweak bremsstrahlung
In the previous subsection we have presented the results of the detailed MonteCarlo compu-tation of neutrino spectra from DM DM annihilations into pairs of SM particles. There are
5
Panels (a) and (e) actually show a small difference in the opposite direction, of which we cannot trace the origin. We tentatively attribute it to small calibration discrepancies among the codes. It is present only at relatively low DM masses.
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10-5 10-4 10-3 10-2 10-1 1 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x Νefrom DM DM®W+W-, MDM= 1000 GeV HaL Geant with EWcorr (a) 10-5 10-4 10-3 10-2 10-1 1 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log xΝe from DM DM®e+e-, MDM= 1000 GeV
HbL
Geant with EWcorr
(b) 10-5 10-4 10-3 10-2 10-1 1 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x ΝΤ from DM DM®ΝΤΝΤ, MDM= 1000 GeV HcL Geant with EWcorr (c)
Figure 3. Comparison of the neutrino spectra with and without the addition of ElectroWeak effects, for a few representative cases.
however higher order effects that can be relevant and are not included in such computations. In general, higher order corrections cannot be computed without having a full DM model: one needs to know the particles mediating the annihilation process in order to include all possible relevant diagrams. But some dominant higher order corrections can be computed in a model-independent way: those enhanced by logarithms of ratios of particle masses, which describe bremsstrahlung. In terms of Feynman diagrams, such process corresponds to at-taching soft or collinear particles to the SM final state particles. While the bremsstrahlung emission due to electromagnetic and strong interactions is automatically performed by Mon-teCarlo codes, the bremsstrahlung emission due to electroweak interactions is not included and is equally significant, if MDM & MW.
We therefore include electroweak bremsstrahlung at leading order in the electroweak couplings by ‘post-processing’ the output of the MonteCarlo as described in [25]. At large DM masses, such bremsstrahlung corrections are enhanced by ln(MDM/MW) logarithms, which
become large for MDM MW. Such enhanced terms are model-independent: in our code we
turn them on abruptly when MDM>∼ MW. In a full DM model, these effects would actually
appear in a smooth model-dependent way when increasing the DM mass. We instead neglect the finite non-logarithmic terms, that cannot be computed in a model-independent way.
In practice, we proceed as follows. In the previous section we computed dNJ →νMC `/dx: the MonteCarlo spectra in x = E/MDM of ν` produced by DM annihilations into a generic
two-body state J . To include EW bremsstrahlung, we convolute such spectra with a set of electroweak splitting functions DEWI→J(z) (probability that I radiates a particle J with energy reduced by a factor z) as follows:
dNI→ν` d ln x (MDM, x) = X J Z 1 x dz DEWI→J(z) dN MC J →ν` d ln x zMDM, x z , (3.9)
The sum is over all EW splittings. The rationale behind this procedure is that splittings are kinematically different from decays and happen before decays, when particles have a large virtuality. The splitting functions are predicted by the SM and listed in [25]. For example, left-handed electrons can radiate neutrinos via the EW splitting eL → νeWT described by
the function DEWeL→νe(z) = δ(1 − z) 1 + α2 4π 3` 2 − `2 2 +α2 4π 1 + z2 1 − z L(1 − z) . (3.10)
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where ` = ln 4MDM2 /MZ2 and L(x) = lnM 2 DMx2 MZ2 + 2 ln 1 + s 1 − M 2 Z MDM2 x2 ! . (3.11)From a phenomenological point of view, such effects can be relevant. Annihilation channels that would produce a low yield of neutrinos (such as DM DM → e+e− or DM DM
→ µ+µ− where muons are severely slowed down or absorbed by matter) are significantly
affected by EW bremsstrahlung because they receive contributions from channels with large yield of neutrinos (such as DM DM → W+W−). This case is shown in panel (b) of figure3.
The other panels in figure 3 show other cases in which the impact of EW effects is sizable. For a channel like W+W− (panel (a)), the radiation leads to a neutrino flux enhanced by about an order of magnitude at low energy. An ‘extreme’ case is reproduced in panel (c): for annihilations directly into tau neutrinos and looking at the flux of ντ themselves, EW
corrections add to the na¨ıve monochromatic spectrum a broad low energy shoulder, as a consequence of the decay of the EW gauge bosons emitted by the primary ντν¯τ pair.
3.3 Spectra at production: results
Figure 4 presents, for reference, an example of our final results for the neutrino spectra at production. We plot the spectra of ν (first column) and ¯ν (second column) for all the channels that we consider for a sample DM mass of 1 TeV. The spectra are normalized per one annihilation of two DM particles. The considered range of x = E/MDM covers from 10−8
to 1. In the third column we zoom on the high energy part, relevant for neutrino detectors such as IceCube.
All these spectra are provided in numerical form.
4 Neutrino propagation
Propagation of neutrinos from the interior of the Sun to the Earth is affected by flavor oscillations and (at energies above tens of GeV) by Neutral Current (NC) and Charged Current (CC) interactions with solar matter, which give rise to absorption and (when a τ lepton is produced) to regeneration of neutrinos with lower energies. In the following we discuss each of these effects and summarize the formalism we employ.
4.1 Formalism
Coherent flavor oscillations and coherence-breaking interactions with matter simultaneously affect neutrino propagation. Following [14–17], the appropriate formalism that marries in a quantum-mechanically consistent way these two aspects, consists in studying the spatial evolution of the 3 × 3 matrix of densities of neutrinos, ρ(Eν), and of anti-neutrinos, ¯ρ(Eν).
The diagonal entries of the density matrix represent the population of the corresponding flavors, whereas the off-diagonal entries quantify the quantum superposition of flavors.6 The
6
Alternatively, the fully numerical approach pursued in Wimpsim [19] consist in writing down an event-based MonteCarlo that follows the path of a single neutrino undergoing oscillations and interactions (with given probabilities). The two approaches yield results which are very well in agreement, for any practical purpose.
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10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x Νe, MDM= 1000 GeV 10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x Νe, MDM= 1000 GeV 10-3 10-2 10-1 1 10-4 10-3 10-2 10-1 1 x=EMDM dN Ν d log x Νe, MDM= 1000 GeV 10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x ΝΜ, MDM= 1000 GeV 10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x ΝΜ, MDM= 1000 GeV 10-3 10-2 10-1 1 10-4 10-3 10-2 10-1 1 x=EMDM dN Ν d log x ΝΜ, MDM= 1000 GeV 10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x ΝΤ, MDM= 1000 GeV 10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x ΝΤ, MDM= 1000 GeV 10-3 10-2 10-1 1 10-4 10-3 10-2 10-1 1 x=EMDM dN Ν d log x ΝΤ, MDM= 1000 GeVFigure 4. Final results for the neutrino spectra at production, including all effects (in particular ElectroWeak corrections). Left column: neu-trino spectra. Central column: antineuneu-trino spectra. Right column: zoom on the high energy portion of the neutrino spectra. Upper row: e flavor; middle row: µ flavor; bottom row: τ flavor.
DM annihilation channel Γ g h Z W t b c q Τ Μ e
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matrices ρ(Eν) and ¯ρ(Eν) satisfy a coupled system of integro-differential equations in thedistance r from the center of the Sun: dρ dr = −i[H, ρ] + dρ dr NC + dρ dr CC (4.1)
with an analogous equation for ¯ρ.
• The first term describes oscillations, computed including the vacuum mixing and the MSW matter effect [53,54]. The effective Hamiltonian reads
H = m †m 2Eν ±√2GF Ne diag (1, 0, 0) − Nn 2 diag (1, 1, 1) , (4.2)
where m is the 3 × 3 neutrino mass matrix, and the + (−) sign applies for neutrinos (anti-neutrinos). One has m†m = V · diag(m21, m22, m23) · V† where m1,2,3 > 0 are the
neutrino masses and V is the neutrino mixing matrix given by
V = R23(θ23) · R13(θ13) · diag (1, eiφ, 1) · R12(θ12) (4.3)
where Rij(θij) represents a rotation by θij in the ij plane and we assume the present
best fit values for the mixing parameters [50–52]7
tan2θsun= 0.45, θatm = 45◦, θ13= 8.8◦,
∆m2sun= 7.5 10−5eV2, |∆m2atm| = 2.45 10−3eV2.
• The second term in eq. (4.1) describes the absorption and re-emission due to NC scat-terings (
ν) N ↔ (
ν)
N∗ (where N is any nucleon in the Sun and with N∗ we denote its possible excited state after the collision), which remove a neutrino from the flux and re-inject it with a lower energy. So they contribute to the evolution equation as:
dρ dr NC = − Z Eν 0 dEν0 dΓNC dE0 ν (Eν, Eν0)ρ(Eν) + Z ∞ Eν dEν0 dΓNC dEν (Eν0, Eν)ρ(Eν0) (4.4) where ΓNC(Eν, Eν0) = Np(r) σ(ν`p → ν`0X) + Nn(r) σ(ν`n → ν`0X). (4.5)
• The third term in eq. (4.1) describes Charged Current (CC) scatterings ( ν)
`N → `±X
of an initial neutrino ν`with energy Eν, which remove the ν` from the flux and produce
a charged lepton ` and scattered hadrons X. They decay back into neutrinos ν`0 and anti-neutrinos ¯ν`0 with lower energy Eν0: their energy distributions are described by the function f`→`0(Eν, E0
ν). When the initial neutrino is ντ (¯ντ), the produced τ− (τ+)
decays promptly before losing energy, giving rise to energetic ντ, ¯νe, ¯νµ (¯ντ, νe, νµ): this
is the tau regeneration phenomenon [46–49].8 When instead the initial neutrino is a νe or νµ we assume that the produced e, µ is totally absorbed and we neglect the
7We neglect the indications possibly in favor of a non-maximal θ
atm and we do not consider the small dependence of the best fit values on the choice of the mass hierarchy.
8
For background information, see also A. Stahl, Physics with τ leptons, Springer Verlag, Heidelberg (2000), where spectra of τ decay products are reviewed. Some analytical approximations can also be found e.g. in [49].
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corresponding low energy neutrinos. CC scatterings thereby affect the propagation of neutrinos with the termdρ dr CC = −{ΓCC, ρ} 2 + Z dEνin Ein ν Πτρτ τ(Eνin)ΓτCC(Eνin)fτ →τ(Eνin, Eν) +Πe,µρ¯τ τ(Eνin)¯ΓτCC(Eνin)fτ →e,µ¯ (Eνin, Eν) , (4.6)
where Π` is the projector on the flavor ν`: e.g. Πe = diag (1, 0, 0). The
matri-ces ΓCC, ¯ΓCC that describe the rates of CC interactions are given by ΓCC(Eν) =
diag (Γe
CC, Γ
µ
CC, ΓτCC), where
Γ`CC = Np(r) σ(ν`p → `X) + Nn(r) σ(ν`n → `X). (4.7)
In both NC and CC processes, we neglect low energy neutrinos that might emerge from the scattered hadrons and light leptons, i.e. the de-excitation of N∗in NC and the decay of X and e, µ in CC. E.g. in particular we neglect the very low energy neutrinos with Eν ∼ mπ,K,µ
coming from the decay at rest of light hadrons/leptons. In order to include them, one should implement neutrino/matter interactions in dedicated codes such as Geant, which currently do not include them. I.e. this would be analogous to the work we performed in 3.1.2, now with neutrinos as primary particles. We postpone this to possible future improvements. We estimate that such a neglected effect would only give a small enhancement in the final flux of neutrinos at very low energy.
4.2 Transition probabilities
We numerically solve the full evolution equation (eq. (4.1)) starting from the initial condition dictated by the spatial distribution of DM annihilations inside the Sun, proportional to n(r)2 given by eq. (2.4). In practice, we numerically computed the full transition probabilities P±(ν`(E0) → νi(E)) from the Sun to the Earth, with ` = {e, µ, τ, ¯e, ¯µ, ¯τ }, i = {1, 2, 3, ¯1, ¯2, ¯3}
and E ≤ E0 in the two cases of normal (P+) and of inverted (P−) neutrino mass hierarchy.
The transition probabilities incorporate all propagation effects. In order to have a qualitative understanding of their behavior, however, we plot them only in the limit E = E0 that excludes the neutrinos produced by regeneration (at E < E0).
Figure5and6show the transition probabilities P (ν`→ ν`0) between flavour eigenstates
of neutrinos (continuous curves) and anti-neutrinos (dashed) from the Sun to the surface of the Earth (before a possible crossing of the Earth, which will be considered in the next subsection). Figure5shows the probabilities computed for θ13= 0. In figure6, θ13is restored
to its actual value θ13= 8.8◦ and therefore we need to distinguish the hierarchy: the upper
row refers to a normal spectrum of neutrinos (m1 m2 m3), while the lower row to an
inverted spectrum (m1 m2≈ m3).
Considering for example P (νe → νe): it goes from 1 −12sin22θ12 at E MeV
(aver-aged vacuum oscillations) to sin2θ12 at larger energies (adiabatic MSW resonance for θ12).
If neutrinos have normal hierarchy, at Eν 100 MeV also θ13 is enhanced by an adiabatic
MSW resonance, and P (νe → νe) drops.9 Many effects happen when Eν ∼ 10 GeV: MSW
resonances cease to be adiabatic, and the solar oscillation wave-length becomes comparable 9
The MSW resonance occurs in a region with width in matter density given by ∆n ≈ 2θn, which is smaller for smaller θ. Numerical solutions to eq. (4.1) obtained using the Runge-Kutta method must thereby employ a smaller step in r when θ13is turned on.
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10-410-310-210-1100101102103104 0.0 0.2 0.4 0.6 0.8 Ν energy in GeV P HΝi ® Νj L Initial Νe®Νe,ΝΜ,ΝΤ 10-410-310-210-1100101102103104 0.0 0.2 0.4 0.6 0.8 Ν energy in GeV P HΝi ® Νj L Initial ΝΜ®Νe,ΝΜ,ΝΤ 10-410-310-210-1100101102103104 0.0 0.2 0.4 0.6 0.8 Ν energy in GeV P HΝi ® Νj L Initial ΝΤ®Νe,ΝΜ,ΝΤAny hierarchy, Θ13=0, no Earth crossing
Figure 5. Neutrino (continuous curves) and anti-neutrino (dashed) transition probability from the Sun to the Earth, assuming θ13= 0. At E 10 GeV the total probability P (νi→Pfνf) is smaller
than 1 because of absorption. Since theta13= 0 in this figure, νµ and ντ are maximally mixed and
their lines overlap. The probabilities plotted here do not include regenerated neutrinos, see text for details. 10-410-310-210-1100101102103104 0.0 0.2 0.4 0.6 0.8 Ν energy in GeV P HΝi ® Νj L Initial Νe®Νe,ΝΜ,ΝΤ 10-410-310-210-1100101102103104 0.0 0.2 0.4 0.6 0.8 Ν energy in GeV P HΝi ® Νj L Initial ΝΜ®Νe,ΝΜ,ΝΤ 10-410-310-210-1100101102103104 0.0 0.2 0.4 0.6 0.8 Ν energy in GeV P HΝi ® Νj L Initial ΝΤ®Νe,ΝΜ,ΝΤ
Normal hierarchy, no Earth crossing
10-410-310-210-1100101102103104 0.0 0.2 0.4 0.6 0.8 Ν energy in GeV P HΝi ® Νj L Initial Νe®Νe,ΝΜ,ΝΤ 10-410-310-210-1100101102103104 0.0 0.2 0.4 0.6 0.8 Ν energy in GeV P HΝi ® Νj L Initial ΝΜ®Νe,ΝΜ,ΝΤ 10-410-310-210-1100101102103104 0.0 0.2 0.4 0.6 0.8 Ν energy in GeV P HΝi ® Νj L Initial ΝΤ®Νe,ΝΜ,ΝΤ
Inverted hierarchy, no Earth crossing
Figure 6. Like figure 5, but allowing for the actual, non-vanishing value of θ13and therefore
distin-guishing the neutrino mass hierarchies.
to the size of the Sun. These two effect cause an increase of P (νe→ νe) towards its vacuum
oscillation value. However this increase gets stopped by neutrino absorption due to interac-tions with solar matter, which causes all probabilities to drop to zero in the limit of large energy. The other oscillation probabilities can be similarly understood.
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10-3 10-2 10-1 1 10 102 103 104 0.0 0.1 0.2 0.3 0.4 0.5 0.6Neutrino energy in GeV
P HΝ1 ,2 ,3 ® Νe,Μ, Τ L
Normal hierarchy, cosJ=-1
P3 Μ P1 Μ P2 Μ 10-3 10-2 10-1 1 10 102 103 104 0.0 0.1 0.2 0.3 0.4 0.5 0.6
Neutrino energy in GeV
P HΝ1 ,2 ,3 ® Νe,Μ, Τ L
Inverted hierarchy, cosJ=-1
P3 Μ
P1 Μ
P2 Μ
Figure 7. Earth crossing oscillation probabilities into νµ (continuous curves) and into ¯νµ (dashed),
for neutrinos crossing vertically the Earth.
Our propagation results agree, in the limit cases, with known results of the past, and in particular with [18] and the works that made use of it. We verified that the unknown neutrino CP-violating phase that enters into oscillations has negligible effects. Furthermore, such results do not depend on the Majorana or Dirac nature of neutrinos.
4.3 Earth crossing
The spectra of neutrinos and anti-neutrinos at the Earth are finally computed (in terms of mass eigenstates νi) by convoluting the transition probabilities with the spectra at
produc-tion as dNν±i dE = X ` Z M E dE0 P±(ν`(E0) → νi(E)) dNνprod` dE0 . (4.8)
The final step consists in taking into account the oscillations in the matter of the Earth. If neutrinos do not cross the Earth, the energy spectra for the neutrino flavour eigenstates are simply given by
dNν±` dEν =X i |V`i|2 dNν±i dEν . (4.9)
If instead neutrinos cross the Earth with zenith angle ϑ (cos ϑ = −1 corresponds to the maximal vertical crossing, and cos ϑ = 0 corresponds to the minimal horizontal crossing), the neutrino fluxes at detection are given by
dNν±` dEν =X i Pearth± (νi→ ν`, Eν, ϑ) dNν±i dEν . (4.10)
where the oscillation probabilities Pearth are readily computed adopting the standard Earth
density model. We neglect neutrino absorption within the Earth (they would be relevant only for energies above ∼ 10 TeV and neutrinos with those energy essentially do not emerge from the Sun, as discussed above).
Some Earth oscillation probabilities are plotted in figure7for illustration; for large and small neutrino energy they approach the limiting values |V`i|2.
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5 Final result
Recapping the results of the previous sections: we provide the energy spectra at detection of neutrino flavor eigenstates dNν±`/dEν produced by one DM annihilation in the Sun, after
taking into account all effects that neutrinos experience during their journey. We provide two sets of spectra: dNν+i/dEν corresponding to neutrinos with normal mass hierarchy, and
dNν−i/dEν corresponding to neutrinos with inverted mass hierarchy.
Figure8presents, for reference, an example of our final results for the neutrino spectra at detection, analogously to figure4. The spectra are, as always, normalized per one annihilation of two DM particles.
In figure9we present a more detailed comparison of the effect of propagation, for a few selected masses and channels. One sees, for instance:
• The effect of flavor vacuum oscillations: for an annihilation into τ+τ− the flux of
electron and muon neutrinos is greatly enhanced and the corresponding flux of tau neutrinos is depleted; for an annihilation into ¯bb, the opposite happens since νe,µmostly
emerge from the b channel.
• The effect of solar matter absorption: moving towards higher masses, the spectra are significantly degraded in energy; the case of the Z spectrum (peaked at production) is the most apparent. For even larger mDM all spectra approach a limit, ‘bell-shaped’
exponential spectrum dictated by the maximum energy to which the Sun is transpar-ent [18].
• The effect of Earth crossing oscillations: the wiggles at around 1 to 10 GeV. All these spectra are provided in numerical form.
The neutrino fluxes can then be converted into fluxes of detectable particles (up-going muons, through-going muons, showers). Various experiments searched for DM neutrinos from the Sun, producing bounds: SuperKamiokande [55], Icecube [56], Antares [57,58] and Baksan [59]. These collaborations report results as bounds on DM annihilations assuming some DM annihilation channel and rates of events (such as through-going-muons) with cuts optimised assuming specific energy spectra. Unfortunately, however, they do not presently report experimental bounds on the neutrino fluxes from the Sun (this kind of analysis could be done assuming monochromatic neutrino spectra at different energies).10 Therefore we cannot compare our improved neutrino fluxes with experimental data and we cannot at present derive improved bounds on all the DM annihilation channels.
6 Conclusions
In this paper we have reconsidered the computation of the high energy neutrino fluxes from the annihilation of DM particles captured in the Sun. With respect to other DM indirect detection search strategies, such a signature is particularly interesting and timely given that: i) It is unique in the sense that it cannot be mimicked by known astrophysical processes; it is therefore virtually background-free at the source, except for solar corona neutrinos [5,6] (the background for detection consists of atmospheric neutrinos). ii) The propagation of neutrinos from the Sun to the Earth is under control, in particular now that the neutrino
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10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x Νe, MDM= 1000 GeV 10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x Νe, MDM= 1000 GeV 10-3 10-2 10-1 1 10-4 10-3 10-2 10-1 1 x=EMDM dN Ν d log x Νe, MDM= 1000 GeV 10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x ΝΜ, MDM= 1000 GeV 10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x ΝΜ, MDM= 1000 GeV 10-3 10-2 10-1 1 10-4 10-3 10-2 10-1 1 x=EMDM dN Ν d log x ΝΜ, MDM= 1000 GeV 10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x ΝΤ, MDM= 1000 GeV 10-810-710-610-510-410-310-210-1 1 10-5 10-4 10-3 10-2 10-1 1 10 102 103 x=EMDM dN Ν d log x ΝΤ, MDM= 1000 GeV 10-3 10-2 10-1 1 10-4 10-3 10-2 10-1 1 x=EMDM dN Ν d log x ΝΤ, MDM= 1000 GeVFigure 8. Final results for the neutrino spectra at detection, including all propagation effects. For definiteness we choose the case of Normal Hierarchy and neutrinos crossing vertically the Earth. Left column: neutrino spectra. Central column: antineutrino spectra. Right column: zoom on the high energy portion of the neutrino spectra. Upper row: e flavor; middle row: µ flavor; bottom row: τ flavor. These plots can be directly compared with those in figure4.
DM annihilation channel Γ g h Z W t b c q Τ Μ e
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1 3 10 30 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Ν energy in GeV dN Ν d log 10 EΝ Νe, MDM= 50 GeV at production at detection, NI at detection, HI b Τ 10 100 3 30 0.0 0.1 0.2 0.3 0.4 Ν energy in GeV dN Ν d log 10 EΝ Νe, MDM= 200 GeV at production at detection, NI at detection, HI b Τ Z 10 30 100 300 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Ν energy in GeV dN Ν d log 10 EΝ Νe, MDM= 600 GeV at production at detection, NI at detection, HI b Τ Z 1 3 10 30 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Ν energy in GeV dN Ν d log 10 EΝ ΝΜ, MDM= 50 GeV at production at detection, NI at detection, HI b Τ 10 100 3 30 0.0 0.1 0.2 0.3 0.4 Ν energy in GeV dN Ν d log 10 EΝ ΝΜ, MDM= 200 GeV at production at detection, NI at detection, HI b Τ Z 10 30 100 300 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Ν energy in GeV dN Ν d log 10 EΝ ΝΜ, MDM= 600 GeV at production at detection, NI at detection, HI b Τ Z 1 3 10 30 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Ν energy in GeV dN Ν d log 10 EΝ ΝΤ, MDM= 50 GeV at production at detection, NI at detection, HI b Τ 10 100 3 30 0.0 0.1 0.2 0.3 0.4 Ν energy in GeV dN Ν d log 10 EΝ ΝΤ, MDM= 200 GeV at production at detection, NI at detection, HI b Τ Z 10 30 100 300 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Ν energy in GeV dN Ν d log 10 EΝ ΝΤ, MDM= 600 GeV at production at detection, NI at detection, HI b Τ ZFigure 9. Comparison of the neutrino spectra at production and detection, showing the effects of propagation. For definiteness we choose the case of neutrinos crossing vertically the Earth. Upper row: e flavor; middle row: µ flavor; bottom row: τ flavor. Different columns: different values of the DM mass.
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oscillation parameters are (almost) all measured and with good precision. iii) The current and upcoming generation of neutrino telescopes are reaching the sensitivity necessary to probe one of the most interesting portions of the parameter space, in competition with other DM search strategies.We have reviewed the capture of DM particles in the Sun, considering the relevant elements in solar matter and different assumptions for the halo velocity distribution.
We have computed the neutrino spectra from the annihilation into all two-body SM annihilation channels, implementing a detailed description of the energy losses of primary particles in solar matter, including secondary neutrinos produced by the scattered matter and low energy ones from the decay at rest of light hadrons/leptons. We performed this computation with the Geant MonteCarlo, after checking against Pythia. We added on top of this the effect of ElectroWeak radiation, which is relevant for DM masses above some hundreds of GeV. We span the range of masses from 5 GeV to 100 TeV.
We then perform the propagation of neutrinos from the center of the Sun to the detector, taking into account (vacuum and matter) oscillations and interaction with solar matter, as well as Earth crossing. We adopt up-to-date oscillation parameters and consider normal or inverted neutrino mass hierarchy.
The main numerical outputs of the computation are given on the PPPC 4 DM ID website (‘DMν’ section). We provide:
. a numerical function for Γann,
. the neutrino spectra at production, including EW radiation effects, . the neutrino spectra at detection, including all propagation effects.
These results can be readily used to derive predictions for the experimental observables.
Acknowledgments
We thank Nicolao Fornengo for useful discussions and a patient cross-check of some of the results of Cirelli et al. in [18]. We thank Paolo Panci, Teresa Montaruli and Sofia Vallecorsa for useful discussions and Eugenio Del Nobile for pointing out a typo.
M.C. acknowledges the hospitality of the Institut d’Astrophysique de Paris (Iap) and of the Theory Unit of Cern where a part of this work was done.
Funding and research infrastructure acknowledgements:
∗ Esf (Estonian Science Foundation) grant MTT8, grant 8499 and SF0690030s09 project, ∗ European Programme PITN-GA-2009-23792 (Unilhc),
∗ European Programme Erdf project 3.2.0304.11-0313 Estonian Scientific Computing Infrastructure (Etais),
∗ European Research Council (Erc) under the EU Seventh Framework Programme (FP7/2007-2013)/Erc Starting Grant (agreement n. 278234 — ‘NewDark’ project) [work of M.C.],
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